If I am correct, Bernoulli's inequality is that $$(1+x)^r \geq 1+rx$$ for all $r\geq 0$ and $x\geq -1$, (*).

Now, I am trying to find a counter-example to show that it doesn't work for negative $r$. I am trying to fix $x\geq -1$ and find a negative $r$ that doesn't satisfy the inequality, but I don't seem to be having much luck.

If I draw the graphs of $(1+x)^r$ and $1+rx$ for $r=-1, r=-2$, I seem to find that there is no positive $x$ and negative $r$ that contradict the statement.

Does $x$ and $r$ have to both not meet their conditions I wrote in (*) for the inequality to fail? Or am I just not looking hard enough?


1 Answer 1


The inequality does work for negative integer $r$, provided $x>-1$ - see the wikipedia page. (In fact the only real number values of $r$ for which it doesn't work are $0<r<1$.) So if you want a counterexample for negative $r$ you have to take $x=-1$.


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